Backpropagation Algorithm Fundamentals
📌 Backpropagation efficiently computes the gradient of the loss function with respect to all network parameters (weights and biases).
🧐 The influence of a weight change differs based on its location: weights in the output layer have a single path effect, while those in hidden layers have multiple paths affecting subsequent neurons and the objective function.
⚙️ Delta variables ($\delta_r^h$), defined as the derivative of the objective function with respect to a neuron's activation, are crucial for avoiding repeated computations during gradient calculation.

Computing Deltas and Gradients
🆕 Deltas for the output layer ($L$) are calculated using the chain rule, involving the derivative of the loss w.r.t. the neuron's output and the derivative of the activation function.
🧮 For hidden layers, the delta of a neuron in layer $l-1$ is calculated by summing the contributions from all affected neurons in the layer above ($l$), weighted by connection weights and the subsequent deltas.
📈 The derivative of the loss function w.r.t. any weight ($w_{ij}^l$) is universally calculated as the product of the delta of the destination neuron and the output of the source neuron: $\frac{\partial L}{\partial w_{ij}^l} = \delta_j^l \cdot a_i^{l-1}$.

Backpropagation Workflow
1️⃣ Forward Stage: Compute the output values ($a$) for all neurons in the network.
2️⃣ Backward Stage (Delta Computation): Compute deltas starting from the output layer backward ($\delta^L$ first), using only the deltas from the layer immediately above to calculate the current layer's deltas.
3️⃣ Gradient Calculation: Once all deltas and output values are available, compute the derivatives w.r.t. all network weights.
🔄 For handling $N$ training examples, this entire forward/backward process must be repeated $N$ times, summing the resulting gradients to get the final gradient for the batch loss function.

Key Points & Insights
➡️ Backpropagation's efficiency stems from computing parameter gradients by moving backward from the output layer, relying only on the computed values of the layer immediately succeeding it.
➡️ The core rule for weight gradient calculation is remarkably consistent across all layers: $\frac{\partial L}{\partial w} = \delta_{\text{destination}} \times \text{Output}_{\text{source}}$.
➡️ Implementing backpropagation involves three main steps: Forward Pass (to get outputs), Backward Pass (to compute deltas), and Gradient Calculation (using deltas and outputs).

📸 Video summarized with SummaryTube.com on Dec 23, 2025, 11:37 UTC